MathematicsChapter 10: Practical Geometry
Sss Construction
SSS means three given sides and no given angle.
Learning objectives
- Explain sss construction in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Definition
SSS means three given sides and no given angle. You draw one side as a base, then from each endpoint you open a compass to the remaining two lengths and draw arcs. The third vertex is where the arcs meet. The triangle is unique if the three lengths can form a triangle at all. That check is the triangle inequality: each pair of sides must sum to more than the third side.
Where / symbols
- three given sides
- SSS means three given sides and no given angle.
- arcs
- You draw one side as a base, then from each endpoint you open a compass to the remaining two lengths and draw arcs.
- unique
- The triangle is unique if the three lengths can form a triangle at all.
- triangle inequality
- That check is the triangle inequality: each pair of sides must sum to more than the third side.
Overview
First test the lengths, then swing the arcs:
Triangle Inequality
For sides a, b, c you need a + b > c, b + c > a, and c + a > b: each pair sum > third. If any pair sums to less than or equal to the third, the arcs never meet in a proper triangle (they miss, or they just kiss on the base). The standard fail example is 3, 4, 8: 3 + 4 = 7, which is less than 8, so no triangle.
Examples 3, 4, 5 works (3+4>5, 3+5>4, 4+5>3) · 3, 4, 8 fails because 3+4 is not greater than 8 · 5, 5, 10 fails: 5+5 = 10, a flat line, not a triangle
How To Draw Sss
Draw the longest side as the base so the third vertex sits clearly above it. Swing arcs from both ends: from one end, arc of radius equal to the second side; from the other end, arc of radius equal to the third side. Their intersection is the third vertex. Join it to both ends. If the lengths are valid, the two arcs meet at exactly one useful point above the base (and a mirror point below, which is the same triangle flipped).
Examples Base 8 cm, then arcs 5 cm and 7 cm from the ends · Use the same compass opening as the written length, not a guess · If the arcs miss, go back and check the inequality
WHAT — Core idea
SSS means three given sides and no given angle. You draw one side as a base, then from each endpoint you open a compass to the remaining two lengths and draw arcs. The third vertex is where the arcs meet. The triangle is unique if the three lengths can form a triangle at all. That check is the triangle inequality: each pair of sides must sum to more than the third side.
HOW — Rules, method, and examples
1. TRIANGLE INEQUALITY: For sides a, b, c you need a + b > c, b + c > a, and c + a > b: each pair sum > third. If any pair sums to less than or equal to the third, the arcs never meet in a proper triangle (they miss, or they just kiss on the base). The standard fail example is 3, 4, 8: 3 + 4 = 7, which is less than 8, so no triangle. 2. HOW TO DRAW SSS: Draw the longest side as the base so the third vertex sits clearly above it. Swing arcs from both ends: from one end, arc of radius equal to the second side; from the other end, arc of radius equal to the third side. Their intersection is the third vertex. Join it to both ends. If the lengths are valid, the two arcs meet at exactly one useful point above the base (and a mirror point below, which is the same triangle flipped).
WHY — Meaning and connections
SSS uniqueness is the reason you can copy a triangle with a compass and straightedge when only sides are known. Do not invent an angle. If the question gives 3, 4, 8, write 'does not satisfy the triangle inequality' instead of forcing a sketch.
HOW — Answer in an exam
SSS: arcs from both ends of the base. Unique when each pair of sides sums to more than the third. 3, 4, 8 fail. Equal sum (5, 5, 10) is also not a triangle. Check inequality before drawing.
Explanation
SSS uniqueness is the reason you can copy a triangle with a compass and straightedge when only sides are known. Do not invent an angle. If the question gives 3, 4, 8, write 'does not satisfy the triangle inequality' instead of forcing a sketch.
In simple words
SSS: arcs from both ends of the base. Unique when each pair of sides sums to more than the third. 3, 4, 8 fail. Equal sum (5, 5, 10) is also not a triangle. Check inequality before drawing.
Key points
- three given sides
- arcs
- unique
- triangle inequality
- each pair sum > third
- 3, 4, 8
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Sss Construction
Sticky Notes · Quick Revision
WHAT
- SSS means three given sides and no given angle.
Key Points
- SSS means three given sides and no given angle.
- For sides a, b, c you need a + b > c, b + c > a, and c + a > b: each pair sum > third.
- Draw the longest side as the base so the third vertex sits clearly above it.
- SSS uniqueness is the reason you can copy a triangle with a compass and straightedge when only sides are known.
HOW
- 1.
- For sides a, b, c you need a + b > c, b + c > a, and c + a > b: each pair sum > third.
- Draw the longest side as the base so the third vertex sits clearly above it.
Examples
- 3, 4, 5 works (3+4>5, 3+5>4, 4+5>3)
- 3, 4, 8 fails because 3+4 is not greater than 8
- 5, 5, 10 fails: 5+5 = 10, a flat line, not a triangle
- Base 8 cm, then arcs 5 cm and 7 cm from the ends
- Use the same compass opening as the written length, not a guess
Common Mistakes
- Correction: SSS: arcs from both ends of the base.
- Correction: Unique when each pair of sides sums to more than the third.
- Correction: 3, 4, 8 fail.
Exam Tip
- SSS: arcs from both ends of the base. Unique when each pair of sides sums to more than the third. 3, 4, 8 fail. Equal sum (5, 5, 10) is also not a triangle. Check inequality before drawing.
60-second revision
- SSS means three given sides and no given angle.
- For sides a, b, c you need a + b > c, b + c > a, and c + a > b: each pair sum > third.
- Draw the longest side as the base so the third vertex sits clearly above it.
- SSS uniqueness is the reason you can copy a triangle with a compass and straightedge when only sides are known.
- SSS: arcs from both ends of the base.
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